![cover image](https://wikiwandv2-19431.kxcdn.com/_next/image?url=https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Donskers_invariance_principle.gif/640px-Donskers_invariance_principle.gif&w=640&q=50)
Donsker's theorem
Statement in probability theory / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Donsker's theorem?
Summarize this article for a 10 year old
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem for empirical distribution functions. Specifically, the theorem states that an appropriately centered and scaled version of the empirical distribution function converges to a Gaussian process.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Donskers_invariance_principle.gif/320px-Donskers_invariance_principle.gif)
Let be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let
. The stochastic process
is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by
The central limit theorem asserts that converges in distribution to a standard Gaussian random variable
as
. Donsker's invariance principle[1][2] extends this convergence to the whole function
. More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space
, the random function
converges in distribution to a standard Brownian motion
as
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/1/18/Donsker_theorem_for_uniform_distributions.gif/640px-Donsker_theorem_for_uniform_distributions.gif)
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/9/94/Donsker_theorem_for_normal_distributions.gif/640px-Donsker_theorem_for_normal_distributions.gif)